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An Improved Multiobjective Particle SwarmOptimization Based on Culture Algorithms

Chunhua Jia * and Hong Zhu

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611371,China; [email protected]* Correspondence: [email protected]; Tel.: +86-133-8816-8253

Academic Editor: Javier Del Ser LorenteReceived: 14 February 2017; Accepted: 18 April 2017; Published: 25 April 2017

Abstract: In this paper, we propose a new approach to raise the performance of multiobjective particleswam optimization. The personal guide and global guide are updated using three kinds of knowledgeextracted from the population based on cultural algorithms. An epsilon domination criterion has beenemployed to enhance the convergence and diversity of the approximate Pareto front. Moreover, asimple polynomial mutation operator has been applied to both the population and the non-dominatedarchive. Experiments on two series of bench test suites have shown the effectiveness of the proposedapproach. A comparison with several other algorithms that are considered good representatives ofparticle swarm optimization solutions has also been conducted, in order to verify the competitiveperformance of the proposed algorithm in solve multiobjective optimization problems.

Keywords: particle swam optimization; cultural algorithm; multiobjective optimization

1. Introduction

Since many real-life problems are in fact multiobjective optimization problems (MOPs), evolutionaryalgorithms used to solve MOPs have attracted much attention in recent years. Multiobjectiveoptimization evolutionary algorithms (MOEAs) mainly include two branches: dominance-based anddecomposition-based [1]. Nondominated sorting genetic algorithm II (NSGA-II) [2], strength Paretoevolutionary algorithm 2 (SPEA2) [3] and multiobjective particle swarm optimization (MOPSO) [4,5]are in the first branch; multiobjective evolutionary algorithm based on decomposition (MOEA/D) [6,7]is in the second branch. For those dominance-based algorithms, the main goals of multiobjectiveoptimization are that the non-dominated set obtained by an algorithm is a good approximation to thetrue Pareto front, and that this non-dominated set is uniformly distributed on the Pareto front [8].

Particle swarm optimization (PSO) is an evolutionary computation technique based on the socialbehavior of species, such as a flock of birds or a school of fish.

To handle MOPs, two problems need to be taken into account. The determination of personal andglobal guides is the first, and second is to maintain the diversity of the population.

Some efforts have been made in selecting the global and local guides. Carlos introduced thesecondary repository that is based on Pareto optimality to store the non-dominated particles in MOPSO,and the global guide is selected using the adaptive grid [4]. In crowding distance (CD)-MOPSO, theglobal guide is selected using crowding distance in MOPSO [9]. In clustering MOPSO, the globalguide is selected based on the clustering technique [10]. Mostaghim and Teich [11] selected the localguide based on the sigma method, while Branke and Mostaghim [12] compared the results of differentmethods in selecting the local guide.

Coello proposed the use of a cultural algorithm to deal with multiobjective optimization problems,in which the knowledge based on the cultural algorithm will affect the selection of both the global guide

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Algorithms 2017, 10, 46 2 of 11

and the local guide [13]. Cultural multiobjective quantum particle swarm optimization (MOQPSO) [1],Cultural MOPSO (CMOPSO) [14], constrained CMOPSO [15], and Cultural dynamic PSO (DPSO ) [16]are all based on a cultural evolution mechanism.

A special domination principle and genetic operator have been used to raise the diversity ofthe Pareto front. Laumanns defined ε-box dominance to combine convergence and diversity [17],while Zitzler introduced the elitism mechanism that performs crossover and mutation on individualsselected from the union of population and repository [18]. Simulated Binary Crossover (SBX) has beenused in the work of Tang [19] and Yu [20]. The Borg MOEA represents a class of algorithms whoseoperators are adaptively selected based on the problem [21]. Diversity-loss-based selection method isused in [22]. Bare-bones (BB)-MOPSO use an approach based on particle diversity to update the globalparticle leaders [23]. Pareto entropy MOPSO (peMOPSO) using cell distance based individual densityto select the global guide [24].

Although there has been considerable research conducted on PSO proposed to solve MOPs, toraise the convergence and diversity of the approximate Pareto front further still remains an issue thatneeds to be considered.

An improved multiobjective particle swam optimization has been developed in this paper.Situational, normative and topographical knowledge are to be used to update the personal andglobal guides. A polynomial mutation operator is applied to every particle in the swarm and in therepository to enhance the convergence and diversity of the Pareto front. Epsilon domination principledepends both on the situation in the adaptive grid and the objective value that has been used.

The paper is structured as follows. The concept of multiobjective problems, cultural algorithms,and particle swarm optimization are briefly reviewed in the next section. In Section 3, a newmultiobjective particle swarm optimization is proposed to improve the convergence and spread to thetrue Pareto front. In Section 4, the results of the experiments and analysis are shown. Conclusions aregiven in Section 5.

2. Preliminary

2.1. Cultural Algorithms

A cultural algorithm usually includes five kinds of knowledge: situational, normative,topographical, domain, and historical knowledge [25].

Situational knowledge is a set of useful instances for the interpretation of the experiences of allindividuals. Situational knowledge guides individuals to move toward exemplars.

Normative knowledge consists of a set of promising ranges of variables. It provides guidelinesfor individual adjustments. Normative knowledge leads individuals to plunge into a good range.

Topographical knowledge divides the whole functional landscape into cells representing differentspatial characteristics, and each cell records the best individual in its specific ranges. Topographicalknowledge leads individuals to catch up to the best cell.

Domain knowledge adopts information about the problem domain to lead the search. Domainknowledge is useful during the search process.

Historical knowledge keeps track of the history of the search process and records key events inthe search. It might be either a considerable move in the search space or a discovery of a landscapechange. Individuals use historical knowledge for guidance in selecting a direction in which to move.

Particles are evaluated by a performance function in each generation. An acceptance functiondetermines which individuals will be used to update the belief space. Experiences of those chosenindividuals are then added to the belief space by the update function. After that, knowledge fromthe belief space influences the selection of individuals for the next generation through the influencefunction. The framework of a cultural algorithm is shown in Figure 1.

Algorithms 2017, 10, 46 3 of 11Algorithms 2017, 10, 46 3 of 11

Figure 1. The framework of a cultural algorithm.

2.2. Multiobjective Optimization Problems

Multiobjective problems (MOPs) are problems whose goal is to optimize multiple objective functions simultaneously. A MOP global minimum problem with n objectives is defined as:

min f(x) = [f1(x), f2(x), …, fn(x)]

s.t.g(x) ≤ 0, h(x) = 0 (1)

where g(x) ≤ 0 and h(x) = 0 represent the constraints.

Definition 1. (Pareto dominance) [26]: x is said to dominate y (denoted as x < y) if "i, fi(x) ≤ fi(y), $ i, fi(x) < fi(y).

Definition 2. (Pareto optimal set) [26]: P* = x $y∈Ω, f(y) f(x).

Definition 3. (Pareto front) [26]: PF* = f(x) = [f1(x), f2(x), …, fn(x)]|x∈P*.

Definition 4. (ε-Pareto dominance) [17]: x is said to ε-dominate y, denoted as x εy, if f(x)·(1 + ε) f(y).

2.3. Particle Swam Optimization

The standard PSO [27] is:

v ( 1)kid+ = ω·v ( )k

id + c1·r1·(p ( )kid – x ( )k

id ) + c2·r2·(p ( )kgd – x ( )k

id )

x ( 1)kid+ = x ( )k

id + v ( 1)kid+

(2)

where v ( )kid and x ( )k

id are the velocity and position of the d-th dimension of the i-th particle in the k-th iteration, respectively, ω is inertia weight, c1 and c2 are the acceleration coefficients. Parameters

r1 and r2 are uniformly distributed random numbers within the range [0, 1]. The parameters p ( )kid and

p ( )kgd indicate the best position that the i-th particle has found so far, and the best position that the

global swarm has found so far. The procedure of PSO is shown in Algorithm 1.

Algorithm 1. The Procedure of Particle Swarm Optimization (PSO) (1) Calculate the fitness function of each particle.

(2) Update p ( )kid and p ( )k

gd .

(3) Update the velocity and position of each particle.

(4) If the stop criterion is not met, go to (1), else p ( )kg is the best position

Figure 1. The framework of a cultural algorithm.

2.2. Multiobjective Optimization Problems

Multiobjective problems (MOPs) are problems whose goal is to optimize multiple objectivefunctions simultaneously. A MOP global minimum problem with n objectives is defined as:

min f(x) = [f1(x), f2(x), . . . , fn(x)]s.t.g(x) ≤ 0, h(x) = 0

(1)

where g(x) ≤ 0 and h(x) = 0 represent the constraints.

Definition 1. (Pareto dominance) [26]: x is said to dominate y (denoted as x < y) if ∀i, fi(x) ≤ fi(y), ∃i, fi(x) < fi(y).

Definition 2. (Pareto optimal set) [26]: P* = x|¬∃y ∈ Ω, f(y) ≺ f(x).

Definition 3. (Pareto front) [26]: PF* = f(x) = [f1(x), f2(x), . . . , fn(x)]|x∈P*.

Definition 4. (ε-Pareto dominance) [17]: x is said to ε-dominate y, denoted as x ≺ εy, if f(x) · (1 + ε) ≺ f(y).

2.3. Particle Swam Optimization

The standard PSO [27] is:

v(k+1)id = ω·v(k)

id + c1·r1

(p(k)

id − x(k)id

)+ c2·r2

(p(k)

gd − x(k)id

)x(k+1)

id = x(k)id + v(k+1)id

(2)

where v(k)id and x(k)id are the velocity and position of the d-th dimension of the i-th particle in the k-th

iteration, respectively,ω is inertia weight, c1 and c2 are the acceleration coefficients. Parameters r1 andr2 are uniformly distributed random numbers within the range [0, 1]. The parameters p(k)

id and p(k)gd

indicate the best position that the i-th particle has found so far, and the best position that the globalswarm has found so far. The procedure of PSO is shown in Algorithm 1.

Algorithm 1. The Procedure of Particle Swarm Optimization (PSO)

(1) Calculate the fitness function of each particle.

(2) Update p(k)id and p(k)

gd .

(3) Update the velocity and position of each particle.

(4) If the stop criterion is not met, go to (1), else p(k)g is the best position

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3. The Proposed Algorithm

3.1. Three Kinds of Knowledge Based on Population

Situational knowledge is determined by all of the personal guide positions. Figure 2 gives thesituational knowledge of a space with two objectives. The obtained local area is split into n partsidentically in each dimension. Situational knowledge is used to generate the personal guide foreach particle.

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Situational knowledge is determined by all of the personal guide positions. Figure 2 gives the situational knowledge of a space with two objectives. The obtained local area is split into n parts identically in each dimension. Situational knowledge is used to generate the personal guide for each particle.

Figure 2. Situational knowledge.

Normative knowledge is based on the minimum and maximum values of non-dominated particles. Normative knowledge of a space with two objectives is shown in Figure 3. The normative knowledge in Figure 3 is updated from L1, U1, L2, U2 to L1’, U1’, L2’, U2’.

Li = min fi(x), Ui = max fi(x) (3)

The normative space is later used to obtain the global guide of the MOPSO by establishing the topographical knowledge.

Figure 3. Normative knowledge.

In order to construct the topographical knowledge, the normative knowledge is employed and then the space is divided into grids, in which each of the resultant hypercubes will then be characterized by all of the lower and upper limits of the hypercubes, as well as the number of non-dominated individuals that are located on that hypercube.

Figure 4 shows the topographical knowledge of a two-objective space. With each generation, the topographic knowledge will be updated. Topographical knowledge is used to find the global guide.


Normative knowledge is based on the minimum and maximum values of non-dominated particles.Normative knowledge of a space with two objectives is shown in Figure 3. The normative knowledgein Figure 3 is updated from L1, U1, L2, U2 to L1’, U1’, L2’, U2’.


The normative space is later used to obtain the global guide of the MOPSO by establishing thetopographical knowledge.

Algorithms 2017, 10, 46 4 of 11



Situational knowledge is determined by all of the personal guide positions. Figure 2 gives the situational knowledge of a space with two objectives. The obtained local area is split into n parts identically in each dimension. Situational knowledge is used to generate the personal guide for each particle.


Normative knowledge is based on the minimum and maximum values of non-dominated particles. Normative knowledge of a space with two objectives is shown in Figure 3. The normative knowledge in Figure 3 is updated from L1, U1, L2, U2 to L1’, U1’, L2’, U2’.


The normative space is later used to obtain the global guide of the MOPSO by establishing the topographical knowledge.


In order to construct the topographical knowledge, the normative knowledge is employed and then the space is divided into grids, in which each of the resultant hypercubes will then be characterized by all of the lower and upper limits of the hypercubes, as well as the number of non-dominated individuals that are located on that hypercube.

Figure 4 shows the topographical knowledge of a two-objective space. With each generation, the topographic knowledge will be updated. Topographical knowledge is used to find the global guide.


In order to construct the topographical knowledge, the normative knowledge is employed andthen the space is divided into grids, in which each of the resultant hypercubes will then be characterized

Algorithms 2017, 10, 46 5 of 11

by all of the lower and upper limits of the hypercubes, as well as the number of non-dominatedindividuals that are located on that hypercube.

Figure 4 shows the topographical knowledge of a two-objective space. With each generation, thetopographic knowledge will be updated. Topographical knowledge is used to find the global guide.Algorithms 2017, 10, 46 5 of 11

Figure 4. Topographical knowledge.

3.2. Personal Guide

As in cultural MOQPSO [1], one dimension of the position of one particle is substituted with a random number s generated in the band, decided by the personal guide of whole population. If the new one dominates the old one, then the new one is employed; otherwise, the old one continues to be used. The detailed procedure of a personal guide is shown in Algorithm 2.

Algorithm 2. The Detailed Procedure of A Personal Guide For the d-th dimension of the i-th particle, (1) bn+1 = max(pbestid|i = 1, 2, …, n);

b1 = min(pbestid|i = 1, 2, …, n); bi = b1 + (bn+1 – b1)·(i – 1)/n, i = 2, 3, …, n

(2) si = rand(bi, bi+1);

(3) Replace x di with si and get a new particle

(4) If the newly generated particle dominates xi, then it replaces xi; break; if not, the newly generated particle is discarded

3.3. Global Guide

Making use of topographical knowledge, a hypercube is selected based on Roulette Wheel Selection, then one member of this hypercube is randomly selected to be the global guide. The probability of selecting one hypercube is inversely proportional to the exponential of the number of non-dominated particles in that hypercube, and the exponential parameter is used here to increase the probability of selecting the hypercube that has less non-dominated particles in it.

3.4. Dominate Criteria Based on the Hypercube

Inferred from [17], a new domination criterion is proposed. The location of each pair of members in the non-dominated archive is used together with the objective value to determine the dominance. The hypercubes that are dominated by hypercube 3, 2 are shown in gray in Figure 5.

Figure 4. Topographical knowledge.

3.2. Personal Guide

As in cultural MOQPSO [1], one dimension of the position of one particle is substituted with arandom number s generated in the band, decided by the personal guide of whole population. If thenew one dominates the old one, then the new one is employed; otherwise, the old one continues to beused. The detailed procedure of a personal guide is shown in Algorithm 2.

Algorithm 2. The Detailed Procedure of A Personal Guide

For the d-th dimension of the i-th particle,(1) bn+1 = max(pbestid|i = 1, 2, . . . , n);

b1 = min(pbestid|i = 1, 2, . . . , n);bi = b1 + (bn+1 – b1)·(i – 1)/n, i = 2, 3, . . . , n

(2) si = rand(bi, bi+1);(3) Replace xd

i with si and get a new particle(4) If the newly generated particle dominates xi, then it replaces xi; break;

if not, the newly generated particle is discarded

3.3. Global Guide

Making use of topographical knowledge, a hypercube is selected based on Roulette WheelSelection, then one member of this hypercube is randomly selected to be the global guide.The probability of selecting one hypercube is inversely proportional to the exponential of the numberof non-dominated particles in that hypercube, and the exponential parameter is used here to increasethe probability of selecting the hypercube that has less non-dominated particles in it.

3.4. Dominate Criteria Based on the Hypercube

Inferred from [17], a new domination criterion is proposed. The location of each pair of membersin the non-dominated archive is used together with the objective value to determine the dominance.The hypercubes that are dominated by hypercube 3, 2 are shown in gray in Figure 5.

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Figure 5. Hypercube.

3.5. Mutation Operator

A polynomial mutation is applied to each particle, and thereafter the non-dominated items produced are added into the repository. A polynomial mutation is applied to every member of the repository, the produced items are added into the repository, and then the repository based on dominates is updated. When the resulting offspring is non-dominated by the individuals in the repository, it is added into the repository. If any individual in the repository dominates the offspring, then the offspring is discarded.

The polynomial mutation is stated as [28]:

sm = s + α·βj (4)

where α = ( ) ( )

( )( ) ( )

1 1

1/ 1

2 1, 0.5

1 2 1 ,otherwise

p

p

v v

v

+

+

− <

− −, β = uj – lj, p is a positive real number, v is a uniformly

distributed random number in [0, 1], and βj is the range of j-th dimension.

3.6. Updating the Repository

Inferred from [29], in which the archive is updated based on the convergence and diversity metric, the archive is updated based on crowding distance in this paper. When the number of non-dominated solutions exceeds the limit size of the repository, first the crowding distance of every item in the archive is computed, and then the one that has the minimum crowding distance is deleted. This is repeated until the number of non-dominated solutions is equal to the limit size of the repository. The update function of the dominance archive is shown in Algorithm 3.

Algorithm 3. The Update Function of the Dominance ArchiveSuppose element f’ is a new element that is to be judged whether to enter the repository (rep). (1) Input rep, f (2) If ¬$f’Î rep:hypercube(f) hypercube(f’) then rep = rep∪f\E (3) If ∃f’:hypercube(f’) = hypercube(f)∩f ε + f’ then rep = rep∪f\f’ (4) If ¬$f’:hypercube(f’) = hypercube(f)∪hypercube(f’) hypercube(f), then rep = rep∪f (5) Output rep

4. Experiments

The proposed approach is compared to the following several multiobjective evolutionary algorithms to test its performance: MOPSO [4], cultural MOPSO [14], cultural quantum-behaved

Figure 5. Hypercube.

3.5. Mutation Operator

A polynomial mutation is applied to each particle, and thereafter the non-dominated itemsproduced are added into the repository. A polynomial mutation is applied to every member ofthe repository, the produced items are added into the repository, and then the repository based ondominates is updated. When the resulting offspring is non-dominated by the individuals in therepository, it is added into the repository. If any individual in the repository dominates the offspring,then the offspring is discarded.

The polynomial mutation is stated as [28]:

sm = s + α·βj (4)

where α =

(2v)1/(p+1) − 1, v < 0.5

1− (2(1− v))1/(p+1), otherwise, β = uj – lj, p is a positive real number, v is a uniformly

distributed random number in [0, 1], and βj is the range of j-th dimension.

3.6. Updating the Repository

Inferred from [29], in which the archive is updated based on the convergence and diversity metric,the archive is updated based on crowding distance in this paper. When the number of non-dominatedsolutions exceeds the limit size of the repository, first the crowding distance of every item in the archiveis computed, and then the one that has the minimum crowding distance is deleted. This is repeateduntil the number of non-dominated solutions is equal to the limit size of the repository. The updatefunction of the dominance archive is shown in Algorithm 3.

Algorithm 3. The Update Function of the Dominance Archive

Suppose element f’ is a new element that is to be judged whether to enter the repository (rep).(1) Input rep, f(2) If ¬∃f’∈ rep:hypercube(f)≺hypercube(f’) then rep = rep∪fE(3) If ∃f’:hypercube(f’) = hypercube(f)∩f≺ε + f’ then rep = rep∪ff’(4) If ¬∃f’:hypercube(f’) = hypercube(f)∪hypercube(f’)≺hypercube(f), then rep = rep∪f(5) Output rep

4. Experiments

The proposed approach is compared to the following several multiobjective evolutionaryalgorithms to test its performance: MOPSO [4], cultural MOPSO [14], cultural quantum-behaved

Algorithms 2017, 10, 46 7 of 11

MOPSO [1], NSGA-II [2], and crowding distance-based MOPSO [9]. The implementation of somealgorithms is available at http://delta.cs.cinvestav.mx/~ccoello/EMOO/EMOOsoftware.html.

ZDT and DTLZ test functions were taken to verify our approach [26]. The number of objectives inthe ZDT test function is 2 and the number of objectives in the DTLZ test function is 3. The number ofvariables is 30 for ZDT1-3, 10 for ZDT4, 10 for ZDT6, 10 for DTLZ1-6 and 20 for DTLZ7.

The algorithm is run for a population size of 100, a repository size of 100 particles, and 30 divisionsfor the adaptive grid in each dimension of the objective space. The number of fitness functionevaluations is 30,000. The results that obtained over 30 independent runs are reported in the following.The parameter of the PSO is time-varied as below [30,31]:

ω = 0.9 - (ita/max_ita) × (0.9 - 0.4) (5)

c1 = 2.5 - (ita/max_ita) × (2.5 - 0.5) (6)

c2 = 0.5 - (ita/max_ita) × (0.9 - 0.4) (7)

4.1. Quantitative Performance Evaluations

Hypervolume is adopted as a quantitative measure in this article, as it is a measure of theperformance in both convergence and diversity [32]. Hypervolume is defined mathematicallyas follows:

Hv = ( ∪p∈PFix|p < x < r) (8)

where ˆ is the Lebesgue measure, and r ∈ Ri is a reference vector that is dominated by all of the validcandidate solutions in PFi.

Table 1 shows the mean and standard deviation values of hypervolume for over 30 independentruns obtained by all of the six algorithms. For further analysis, the T test is performed to evaluatethe significant difference between two independent samples, and the results are illustrated in Table 1,where the value of α is set to 0.05. The best result of one test function with different methods is shownin bold. It can be seen that the proposed improved cultural MOPSO (ICMOPSO) has achieved the bestperformance in hypervolume for nine of the 12 test functions, and it is superior to MOPSO for all 12 ofthe test functions, superior to Cultural MOPSO except in ZDT6, superior to NSGA-II except in DTLZ5,superior to CD-MOPSO except in five functions, and finally inferior to CD-MOPSO only in DTLZ2.

Table 1. Hypervolume values for 30 runs of the test functions.

Test Functions MOPSO ICMOPSO

CulturalMOQPSO

CulturalMOPSO NSGA-II CD

MOPSO

ZDT1 mean 5.56 × 10−1 8.73 × 10−1 8.42 × 10−1 8.66 × 10−1 8.11 × 10−1 8.74 × 10−1

standard deviation (std) 9.03 × 10−2 5.38 × 10−3 8.01 × 10−2 7.00 × 10−3 8.46 × 10−3 4.76 × 10−3

p-value 8.42 × 10−2 3.72 × 10−2 8.82 × 10−5 6.65 × 10−40 6.25 × 10−1

ZDT2 mean 2.88 × 10−1 5.38 × 10−1 4.72 × 10−1 4.20 × 10−1 1.20 × 10−1 5.37 × 10−1

std 1.19 × 10−1 7.88 × 10−3 1.07 × 10−1 1.91 × 10−1 2.22 × 10−2 6.43 × 10−3

p-value 1.69 × 10−16 1.40 × 10−3 1.31 × 10−5 6.53 × 10−66 8.25 × 10−1

ZDT3 mean 1.00 1.01 9.39 × 10−1 7.61 × 10−1 7.60 × 10−1 1.01std 5.17 × 10−3 4.01 × 10-3 1.43 × 10−1 4.00 × 10−1 1.31 × 10−2 4.37 × 10−3

p-value 1.72 × 10−7 0.0110 1.35 × 10−3 2.35 × 10−66 8.76 × 10−1

ZDT4 mean 0.00 8.71 × 10−1 8.67 × 10-1 0.00 3.66 × 10−1 0.00std 0.00 6.08 × 10−3 5.18 × 10−3 0.00 1.77 × 10−1 0.00

p-value 1.80 × 10−118 1.59 × 10−2 1.80 × 10−118 1.80 × 10−118 1.80 × 10−118

ZDT6 mean 4.94 × 10−1 5.04 × 10−1 4.65 × 10−1 5.03 × 10−1 0.00 5.05 × 10−1

std 7.43 × 10−3 4.88 × 10−3 1.12 × 10−1 7.63 × 10−3 0.00 5.77 × 10−3

p-value 5.97 × 10−08 6.31 × 10−2 8.61 × 10−1 3.40 × 10−110 8.76 × 10−1

DTLZ1 mean 0.00 1.30 1.30 0.00 0.00 0.00std 0.00 2.02 × 10−3 3.07 × 10−3 0.00 0.00 0.00

p-value 2.90 × 10−156 6.44 × 10−1 2.90 × 10−156 2.90 × 10−156 2.90 × 10−156

http://delta.cs.cinvestav.mx/~ccoello/EMOO/EMOOsoftware.html

Algorithms 2017, 10, 46 8 of 11

Table 1. Cont.

Test Functions MOPSO ICMOPSO

CulturalMOQPSO

CulturalMOPSO NSGA-II CD

MOPSO

DTLZ2 mean 6.74 × 10-1 7.15 × 10−1 4.27 × 10−1 2.46 × 10−1 6.93 × 10−1 7.20 × 10−1

std 1.50 × 10−2 9.74 × 10−3 2.41 × 10−2 1.26 × 10−2 8.35 × 10−3 8.12 × 10−3

p-value 4.76 × 10−18 3.83 × 10−54 1.33 × 10−78 3.17 × 10−13 3.85 × 10−2

DTLZ3 mean 0.00 7.16 × 10−1 7.15 × 10−1 0.00 0.00 0.00std 0.00 7.63 × 10−3 9.61 × 10−3 0.00 0.00 0.00

p-value 8.20 × 10−108 9.47 × 10−1 8.20 × 10−108 8.20 × 10−108 8.20 × 10−108

DTLZ4 mean 6.82 × 10−1 7.29 × 10−1 5.46 × 10−1 1.67 × 10−1 6.65 × 10−1 7.22 × 10−1

std 1.56 × 10−2 7.45 × 10−3 5.70 × 10−2 6.76 × 10−2 3.11 × 10−2 8.74 × 10−3

p-value 1.55 × 10−21 9.53 × 10−25 6.11 × 10−47 9.22 × 10−16 8.66 × 10−4

DTLZ5 mean 4.34 × 10−1 4.39 × 10−1 4.35 × 10−1 2.43 × 10−1 4.38 × 10−1 4.38 × 10−1

std 6.21 × 10−3 4.94×10−3 8.00 × 10−3 4.96 × 10−3 5.74 × 10−3 5.86 × 10−3

p-value 1.65 × 10−3 3.72 × 10−2 2.13 × 10−77 5.64 × 10−1 3.14 × 10−1

DTLZ6 mean 4.29 × 10−1 4.40×10−1 4.21 × 10−1 2.36 × 10−1 0.00 4.39 × 10−1

std 6.99 × 10−3 5.97×10−3 7.53 × 10−3 2.81 × 10−2 0.00 6.85 × 10−3

p-value 1.43 × 10−8 5.64 × 10−15 3.58 × 10−43 9.89 × 10−102 6.28 × 10−1

DTLZ7 mean 1.27 1.43 1.21 5.44 × 10−1 1.47 × 10−1 1.38std 2.93 × 10−2 3.20 × 10−2 4.10 × 10−2 8.36 × 10−2 4.43 × 10−2 4.25 × 10−2

p-value 1.19 × 10−28 2.91 × 10−31 1.98 × 10−51 5.57 × 10−73 5.54 × 10−6

Better/similar/worse 12/0/0 9/3/0 11/1/0 11/1/0 5/6/1

4.2. Comparison of the Results Using Hypercube-Based ε-Domination and Normal Domination Hypervolume

To demonstrate the effectiveness of the hypercube-based ε-domination criteria introduced in thispaper, DTLZ1 and DTLZ7 are chosen as examples.

Figures 6 and 7 show the Pareto fronts obtained using two kinds of dominate criterions for DTLZ1and DTLZ7, respectively. It can be observed that the approximate Pareto fronts found by the improvedCultural MOPSO using hypercube-based ε-domination are more well-distributed.

The adjacent members of the repository may be deleted based on the hypercube-based dominationcriterion, because they may locate in the same grid which is not expected in scenes like ZDT4, ZDT6and DTLZ6. So, for ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, DTLZ5, and DTLZ6, normal dominationcriterion should be used.

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p-value 5.97 × 10−08 6.31 × 10−2 8.61 × 10−1 3.40 × 10−110 8.76 × 10−1 DTLZ1 mean 0.00 1.30 1.30 0.00 0.00 0.00

std 0.00 2.02 × 10−3 3.07 × 10−3 0.00 0.00 0.00 p-value 2.90 × 10−156 6.44 × 10−1 2.90 × 10−156 2.90 × 10−156 2.90 × 10−156

DTLZ2 mean 6.74 × 10-1 7.15 × 10−1 4.27 × 10−1 2.46 × 10−1 6.93 × 10−1 7.20 × 10−1 std 1.50 × 10−2 9.74 × 10−3 2.41 × 10−2 1.26 × 10−2 8.35 × 10−3 8.12 × 10−3 p-value 4.76 × 10−18 3.83 × 10−54 1.33 × 10−78 3.17 × 10−13 3.85 × 10−2

DTLZ3 mean 0.00 7.16 × 10−1 7.15 × 10−1 0.00 0.00 0.00 std 0.00 7.63 × 10−3 9.61 × 10−3 0.00 0.00 0.00 p-value 8.20 × 10−108 9.47 × 10−1 8.20 × 10−108 8.20 × 10−108 8.20 × 10−108

DTLZ4 mean 6.82 × 10−1 7.29 × 10−1 5.46 × 10−1 1.67 × 10−1 6.65 × 10−1 7.22 × 10−1 std 1.56 × 10−2 7.45 × 10−3 5.70 × 10−2 6.76 × 10−2 3.11 × 10−2 8.74 × 10−3 p-value 1.55 × 10−21 9.53 × 10−25 6.11 × 10−47 9.22 × 10−16 8.66 × 10−4

DTLZ5 mean 4.34 × 10−1 4.39 × 10−1 4.35 × 10−1 2.43 × 10−1 4.38 × 10−1 4.38 × 10−1 std 6.21 × 10−3 4.94×10−3 8.00 × 10−3 4.96 × 10−3 5.74 × 10−3 5.86 × 10−3 p-value 1.65 × 10−3 3.72 × 10−2 2.13 × 10−77 5.64 × 10−1 3.14 × 10−1

DTLZ6 mean 4.29 × 10−1 4.40×10−1 4.21 × 10−1 2.36 × 10−1 0.00 4.39 × 10−1 std 6.99 × 10−3 5.97×10−3 7.53 × 10−3 2.81 × 10−2 0.00 6.85 × 10−3 p-value 1.43 × 10−8 5.64 × 10−15 3.58 × 10−43 9.89 × 10−102 6.28 × 10−1

DTLZ7 mean 1.27 1.43 1.21 5.44 × 10−1 1.47 × 10−1 1.38 std 2.93 × 10−2 3.20 × 10−2 4.10 × 10−2 8.36 × 10−2 4.43 × 10−2 4.25 × 10−2 p-value 1.19 × 10−28 2.91 × 10−31 1.98 × 10−51 5.57 × 10−73 5.54 × 10−6

Better/similar/worse 12/0/0 9/3/0 11/1/0 11/1/0 5/6/1

4.2. Comparison of the Results Using Hypercube-Based ε -Domination and Normal Domination Hypervolume

To demonstrate the effectiveness of the hypercube-based ε-domination criteria introduced in this paper, DTLZ1 and DTLZ7 are chosen as examples.

Figures 6 and 7 show the Pareto fronts obtained using two kinds of dominate criterions for DTLZ1 and DTLZ7, respectively. It can be observed that the approximate Pareto fronts found by the improved Cultural MOPSO using hypercube-based ε-domination are more well-distributed.

The adjacent members of the repository may be deleted based on the hypercube-based domination criterion, because they may locate in the same grid which is not expected in scenes like ZDT4, ZDT6 and DTLZ6. So, for ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, DTLZ5, and DTLZ6, normal domination criterion should be used.

(a) (b)

Figure 6. Pareto fronts for DTLZ1: (a) the Pareto fronts obtained using normal domination criterion; (b) the Pareto fronts obtained using hypercube-based ε-domination. Figure 6. Pareto fronts for DTLZ1: (a) the Pareto fronts obtained using normal domination criterion;(b) the Pareto fronts obtained using hypercube-based ε-domination.

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(a) (b)

Figure 7. Pareto fronts for DTLZ7: (a) the Pareto fronts obtained using normal domination criterion; (b) the Pareto fronts obtained using hypercube-based ε-domination.

4.3. Comparison of the Result with and without the Mutation Operator

The comparison of the Pareto fronts of ZDT1 and ZDT3 with and without the mutation operator is shown in Figures 8 and 9. It can be seen that using the mutation operator can result in a more well-distributed approximation of the Pareto front.

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Figure 8. Pareto fronts for ZDT1: (a) the Pareto fronts obtained without using the mutation operator; (b) the Pareto fronts obtained using the mutation operator.

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Figure 7. Pareto fronts for DTLZ7: (a) the Pareto fronts obtained using normal domination criterion;(b) the Pareto fronts obtained using hypercube-based ε-domination.


The comparison of the Pareto fronts of ZDT1 and ZDT3 with and without the mutation operatoris shown in Figures 8 and 9. It can be seen that using the mutation operator can result in a morewell-distributed approximation of the Pareto front.

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Figure 8. Pareto fronts for ZDT1: (a) the Pareto fronts obtained without using the mutation operator;(b) the Pareto fronts obtained using the mutation operator.

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Figure 9. Pareto fronts for ZDT3: (a) the Pareto fronts obtained without using the mutation operator;(b) the Pareto fronts obtained using the mutation operator.

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5. Conclusions

A new approach has been proposed in this paper to improve the performance of MOPSO.It updates the personal and global guides based on three kinds of knowledge that are extracted from acultural algorithm. An epsilon domination criterion has been utilized. What is more, a polynomialmutation operator is applied in order to raise the performance of the Pareto front. Experiments ona series of ZDT and DTLZ test functions has been conducted to compare the proposed method withseveral state-of-the-art MOPSO algorithms. The results indicate that the proposed approach is a viablealternative, since its average performance is highly competitive. The performance of the proposedalgorithm in dynamic optimization problems merits further study in future works.

Acknowledgments: The research work was supported by the National Defense Pre-Research Foundation of Chinaunder Grant No. 9140A27020215DZ02001.

Author Contributions: Chunhua Jia and Hong Zhu conceived and designed the experiments; Chunhua Jiaperformed the experiments; Chunhua Jia and Hong Zhu analyzed the data; Hong Zhu contributed analysis tools;Chunhua Jia wrote the paper.

Conflicts of Interest: The authors declare no conflict of interest.

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© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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